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EMPower™ 17

EMPower Product SamplerCorrelations for Geometry

EMPower Mathematics Topic Correlation Guide

Over, Around, and Within: Geometry and Measurement

Book Description: Students explore the features and measures of basic shapes. Perimeter and area of two-dimensional shapes and volume of rectangular solids provide the focus.

Lesson Number:

Lesson Name: Mathematical Concepts/ Topics Covered:

Opening the Unit

Geometry Groundwork

• Shapes identified and sketched • Angles introduced • Geometry vocabulary list started• Prior Geometry knowledge assessed

Lesson 1: Sharing Secret Designs

• Two-dimensional shape characteristics identified

• 12 Basic Geometric Shapes identified and described

Lesson 2: Get It Right • Angles identified and described with conventional notation

• Right angles introduced• Angle measurements estimated with 90˚

benchmark and determined precisely with protractors

Lesson 3: Get It Straight • Straight (180°) angles explored• Sums of angles in triangles and rectangles

established

Lesson 4: Giant-Size • Similar shapes identified and described• Length and width dimensions introduced and

measured• Perimeters determined by adding

Interim Assessment 1

Shapes and Angles • Attributes of shapes’ and angle measurements’ knowledge assessed

Lesson 5: Line Up by Size • Area and perimeter distinguished

Lesson 6: Combining Rectangles

• Rectangle area calculated in square centimeters• Composite shapes’ areas and perimeters

compared

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18 EMPower™ Product Sampler

Lesson 7: Disappearing Gridlines

• Formulas for area and perimeter derived• Missing dimension values determined• Area of a right triangle calculated

Lesson 8: Conversion Experience

• Standard English Units introduced• Linear unit conversions established

Lesson 9: Squarely in English

• Square units – square inches, feet, and yards constructed and connected with area measure

• Square unit conversions established

Lesson 10: Scale Down • Scale drawings made and steps for scaling analyzed

Interim Assessment 2

A Fresh Look • Area, perimeter, measurement, and scale knowledge applied and assessed

Lesson 11: Filling the Room • Volume explored as capacity• Third dimension – height becomes apparent

Lesson 12: Cheese Cubes, Anyone?

• Cubic inch introduced then used to measure volume

• Volume formula derived

Lesson 13: On the Surface • Surface area and volume compared• Surface area and shape relationship

generalized

Closing the Unit

Design a Box • Geometry and measurement knowledge applied and assessed

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Over, Around, and WithinGeometry and Measurement

STUDENT BOOK

TERC

Mary Jane Schmitt, Myriam Steinback,Tricia Donovan, and Martha Merson

Also available:

EMPower™ Opening the Unit: Geometry Groundwork 1

Often when we are making something, ideas associated with the studyof geometry arise. Do you see shapes within shapes? How do parts fittogether? How does measuring help you figure things out when you arecooking, building, or sewing?

In this unit, you will learn about shape, line, and form. You will explorethe skills that allow people to work mathematically with shapes ineveryday life.

Opening the Unit:Geometry

Groundwork

Is there geometry in your life?

2 Over, Around, and Within: Geometry and Measurement EMPower™

Activity 1: Making a Mind Map

Make a Mind Map using words, numbers, pictures, or ideas that come tomind when you think of geometry and measurement.

GEOMETRY

MEASUREMENT


Activity 2: Making an Angle Demonstrator

Make your own angle demonstrator with the parts that your teacher willgive you.

• Cut out the two strips of paper and place one on top of the other sothe two dots are aligned.

• Put a fastener through the two dots so the paper strips can open andclose like a mouth.

Now you have an angle demonstrator. Make another one for home use, ifyou like.

Activity 3: Initial Assessment

Your teacher will show you some problems and ask you to check off howyou feel about your ability to solve each problem:

___ Can do ____ Don’t know how ____ Not sure

Have you ever noticed that every new place you work has its own wordsor specialized vocabulary? This is true of topics in math too. In everylesson you will be introduced to some specialized vocabulary. Do notworry if you see words in the problems that you do not recognize. Youcan write some words down and look them up later, or learn as you go.

Activity 4: Homemade Objects

Choose an object from the collection provided, or use a homemadeobject of your own.

Object: ________________________________________

1. Sketch your object.


Activity 4: Homemade Objects (continued)

2. What shapes make up your object? Make a sketch of each shapeyou see. Name each shape.

Sketch Sheet

3. When you finish drawing and naming the shapes, trade your objectand Sketch Sheet with a classmate. See if you can find anyadditional shapes in your partner’s object. Add them to yourpartner’s Sketch Sheet and name them.

Sketch of Shapes Observed Name or Description of Shapes


Practice: Using Geometry

Write three paragraphs about how you use geometry and measurement.

• In your work…

• At home…

• In everyday life in your community…


Practice: Seeing Geometry

1. Sketch an object that has some interesting shapes.

2. What shapes and angles do you see in the object?

3. What measurements could you take of the object?


Extension: Name the Shapes

The following are traditional Japanese patterns.

1. Sketch and name the shapes you see within this pattern.

2. Sketch and name the shapes you see within this pattern.

3. Many countries have traditional designs that include geometricshapes.

a. Research a design from your own or another tradition.

b. Sketch the design.

c. Name the shapes you see within the design.


Geometry and Measurement Unit Goals

• Recognize and describe shapes and their characteristics.

• Find area and perimeter of rectangles and volume of rectangularsolids.

• Make drawings to scale.

• Use linear, square, and cubic units.

• Use spatial reasoning to solve problems.

• Make generalizations about two- and three-dimensional shapes.

My Own Goals

Over, Around, and WithinGeometry and Measurement

TEACHER BOOK

TERC

Mary Jane Schmitt, Myriam Steinback,Tricia Donovan, and Martha Merson

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Also available:

EMPower™ Facilitating Opening the Unit: Geometry Groundwork 1

Synopsis This session assesses students’ familiarity with basic concepts and vocabularyassociated with geometry and measurement. The activities establish a baseline byextending a variety of opportunities for students to interact with and comment onaspects of shapes, including dimensions, angle size, perimeter, area, and volume.

1. Students complete a Mind Map, recording their ideas and associations withgeometry and measurement.

2. Students complete a self-assessment based on several problems.

3. Students discuss the geometrical shapes in homemade objects.

Objectives • Demonstrate prior knowledge of geometry

• Construct an angle demonstrator and record an angle

• Keep a record of vocabulary useful to the study of geometry

• Identify and sketch shapes

Facilitating Opening the Unit:

GeometryGroundwork

Is there geometry in your life?

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Materials/Prep • Brad fasteners

• Homemade objects and other objects with interesting shapes, such as pictureframes, hand-sewn clothes, baked goods, and musical instruments

• Newsprint

• Paper

• Paper-clip boxes

• Rulers marked with centimeters andinches

• Scissors

• Shoe boxes

Ask students to bring homemade objects oftheir own to class.

Copy angle demonstrators (Blackline Master 1) on heavy paper, and cut out twostrips per student. This adjustable tool allows students to replicate angles ofdifferent sizes.

Make copies of the Initial Assessment, p. 161, one for each student.

Opening DiscussionBegin by asking everyone to estimate sizes with their hands:

How big is a foot? A centimeter?

Look around; do we agree?

Provide an overview of the unit, and set the tone for the class. Tell students:

Measurement is key to exploring ideas of geometry, so we will start there. Inthis class, frequently there will be more than one right answer and many waysto find the answer, yet there also will be times when only one answer will beright.

A centimeter is an example of a standard metric measurement agreed upon bypeople around the world.

Give some examples of the advantages of standard units of measure:

Every city has a Department of Weights and Measures to monitor and enforcestandards. For example, when you buy a gallon of gas, you are assured thatevery pump at every gasoline station will measure an exact gallon.

Nearly everything constructed by human beings began as a drawing. In thisunit, you will learn the geometry useful for planning and designing. You willmeasure, draw, and plan on paper (in two dimensions) and build, estimate, andmeasure solids (in three dimensions).



Activity 1: Making a Mind Map The Mind Map is the first artifact detailing students’ prior knowledge of geometryand measurement. By the end of Opening the Unit, you should have severalartifacts:

• Mind Map (Activity 1)

• Students’ Initial Assessment (Activity 3)

• Labeled sketches from Activity 4: Homemade Objects

• Writing samples from Practice: Using Geometry

Start the Mind Map by writing “Geometry” and “Measurement” on the board.Explain the Mind Map concept:

Mind Maps are an effective way to generate information on a topic when youwant to get ideas down quickly. As you brainstorm, you begin to organize ideasby making connections. Remember: When you are brainstorming, there are nowrong answers.

Ask:

Where do you see measurements in your everyday life? What do you knowabout them?

List some types of measurement units.

What do you know about geometry?

Write down a few ideas. Model clustering them, linking related ideas with lines,and then ask students to record their own ideas on p. 152 of the Student Book.After a few minutes, ask students for examples from their work.

As terms related to geometry and measurement arise, start a class vocabulary list.Do not worry about formal definitions; use what students themselves say. You willwant to add and refer to this word list, so write it on newsprint. Students can takenotes in the Vocabulary section of their books, starting on p. 149.

This is an example of one student’s mind map:



Activity 2: Making an Angle Demonstrator Give every student two strips of paper (from Blackline Master 1) and a bradfastener. Model making an angle demonstrator. Place one strip on top of the otherso the two dots are aligned. Put a fastener through the two dots so the parts canopen and close like a mouth. (Directions are also in the Student Book, p. 3.) Showstudents how to use the angle demonstrator by holding it against an angle (forexample, the corner of the board); trace the inside lines of the angle demonstratoronto a piece of paper. Tell students they will use their angle demonstrators in theInitial Assessment and again in a later lesson.

Activity 3: Initial Assessment Because the Initial Assessment can look intimidating, it is recommended that youshow the tasks on an overhead projector or otherwise display them for the wholeclass to see. In addition, distribute a copy of the Initial Assessment to everystudent.

Then explain that this is a test where students do not solve the problems. Insteadthey are to check off “Can do,”“Don’t know how,” or “Not sure” in response towhether they feel they can solve the problem. Tell your students that the resultswill help you estimate the amount of time they may need on different topicscovered in the unit, such as measurement, area, perimeter, and volume.

Show the tasks on the overhead projector. Remind students not to solve theproblems, but to assess their confidence in their ability to solve them. Students canread silently or you can read to them. Keep the momentum going, reading theproblem and then quickly moving on to the next one.

Collect the assessments and analyze them later.

Heads Up!If most of your class indicate they can do all of the tasks, and their written work on UsingGeometry is of high quality, photocopy the Initial Assessment, and at the next class meetingask students to complete the tasks. See Checklist (p. 7) for information on instructionaldecisions.



Activity 4: Homemade Objects Every student needs an object to complete this activity. There are two options:

• Students can bring homemade objects to class.

• Students can choose an item from your collection.

Refer students to Homemade Objects, Student Book, p. 3. Give them time to sketch,identify, and label shapes within the object. Some objects will lend themselvesbetter to this activity than others. Convey to students that you do not expect theirrepresentations to be perfect.

Model describing how you made one of your own objects. Include geometryterms and the vocabulary of measurement in your description. Then ask studentsto describe the shapes within their objects.

Summary DiscussionList terms students would like to remember or knowmore about based on discussions of objects theymade or the Initial Assessment. If there is time,orient students to their books. They should answerthe Reflection questions (p. 152). Also have themread through Unit Goals (p. 8).

Practice Using Geometry, p. 5This is the initial written assessment for the unit. It will be useful to have a sampleof each student’s work, explaining how he or she sees and uses geometry andmeasurement.

Seeing Geometry, p. 6Students choose another object, sketch it, and describe its geometric propertiesand measurements.

ExtensionName the Shapes, p. 7

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Looking CloselyObserve whether students are able to

Demonstrate prior knowledge of geometryDo students have experience with this area of mathematics? Do they seemcurious?

Construct an angle demonstrator and record an angleConstructing a measurement tool will help students navigate through theactivities of this unit. In addition, it will give you insight into their facility withconstructing, aligning, and tracing—skills they will call on as they find area,perimeter, and volume, and record angles. Take note of students who seem adeptat this activity; later you can assign students to work in pairs in ways that nurtureemerging learners and challenge more advanced students. If some students havedifficulty constructing the tools, pair these students with those who havesucceeded and can offer help.

Keep a record of vocabulary useful to the study of geometryRather than consulting a dictionary, try to piece together definitions based onstudents’ prior knowledge. These definitions can be refined over time.

Do English language learners know geometric terms in their own languages? Askthem to record these as well.

Identify and sketch shapesCan students identify and name the overall shape of objects? Can they identifyand name shapes within objects? Direct students to compare the attributes ofshapes, such as length and position of sides and the number and shape of angles(e.g., wide-open or pointy).



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LESSON COMMENTARY: OPENING THE UNIT

RationaleThis lesson is an invitation for students at all levels to discuss shapes,measurements, and the ways in which geometry and measurement contribute toour ability to construct and describe objects. Constructing a measurement toolhelps students internalize a sense of angle size. EMPower builds on students’ priorknowledge; therefore, objects of students’ own making are an ideal starting point.

Math BackgroundRecognizing the basic shapes—rectangles, squares, circles, and triangles—iscritical to the study of geometry and measurement. Observing the characteristicsof a shape, such as the number of angles or sides, their relative sizes, and the sides’relative positions, allows us to tap into the established knowledge base about theshape, for example, how to find the measurement for a missing side or angle, orthe formula for area.

The world is full of squiggly lines and amorphous, incomplete, or open shapes.Open shapes are problematic when determining area. With no clear boundaries,open shapes lead to questions about infinity. Amorphous shapes present achallenge of a different kind. Finding the area of a shape with curves demandsmaking approximations using rectangles, some of which are tiny, to fill in theshape. In calculus, this is done by finding the area under curve through integrals.However, knowing how to find the area of rectangular, triangular, or hexagonalshapes is the first step toward arriving at an estimate of the area or volume of anamorphous shape.

Facilitation To promote the communication central to mathematics, ask students to intervieweach other about their homemade objects. Post interview questions such as

What steps did you take to make the object?

What geometric shapes are part of your object?

What did you measure?

What were the most difficult and enjoyable parts of making the object?

In some classes, interviewing can be difficult, given the English-proficiency level ofthe students. Further simplify the task by giving sentence starters, e.g., “I have a_____; I made it by ______.”



OPENING THE UNIT IN ACTION

Teachers and students call on their powers of observation.

One teacher’s homemade item sparks discussion that calls on shape names.

Two of my students seemed to wish they had stayed home today, but as theystarted explaining how they made their objects and measured them, theybecame engaged in the activity. I noticed one student did not know how toread a ruler. “Rectangle” was not in his vocabulary; he called the cereal box asquare and measured the sides to see whether they were the same. The otherreluctant student finished drawing his shapes quickly, but the longer he hadto wait for his partner to finish drawing, the more he looked at his object,and the more detail he added to his drawing. By the time these two studentsswitched objects to see what the other had missed, the student who wasinitially resistant was noticing all sorts of things about his partner’s object.He tapped into great powers of observation, and then he suddenly looked atthe ruler he could not read and started asking questions. He actually figuredout 1/4 , 1/2 , and 3/4 inches.

Phyllis FlanaganRock Valley College Adult Education Center, Rockford, IL

The night before I taught this lesson, I decided to make a heart-shaped cakesince it was close to Valentine’s Day. Before I went into class, I was thinkingabout geometry.

I had a box of cake mix, a can of frosting, and square and round pans. I wasfaced with a few design issues. My plan was to cut the round cake in two andattach the two halves to the square as the humps of the heart. So I had tomake sure the side of the square pan was equal to the diameter of the roundpan. That was easy. I chose 8-inch pans.

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continued from previous page

I asked students to speculate: “What do you think I did to make the cake thisshape? What pan(s) do you imagine I used?”

Student: “You used a square pan and cut out a heart.”

(I made a mental note that she called the rectangle that she drew a “square”):

Teacher: “That’s one way, but that’s not how I did it.”

Student: “I think you had a big circular pan and cut off pieces. Like this:”

Teacher: “That’s another way, but not how I did it.”

Student: “You had a heart-shaped pan.”

Teacher: “No, but I’ll give you a hint. I used two pans, a square one and around one. Here they are.”

(Silence.) Then:

Student: “Okay, I see two parts of the circle, and you pushed the squarebetween them.” (He gestured over the cake.)

Teacher: “Yeah, that’s exactly what I did. Who’s hungry?”

Mary Jane SchmittBridge to Learning and Literacy, Harvard University, MA


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